Abstract
Gol’dberg considered the class of functions analytic in the unit disc with unequal positive numbers of zeros and ones there. The maximum modulus of zero- and one-places in this class is non-trivially bounded from below by the universal constant A 2. This constant determines a fundamental limit of controller design in engineering, and has applications when estimating covering regions for composites of fixed point free functions with schlicht functions. The lower bound for A 2 is improved in this note by considering simultaneously the extremal functions f and 1 — f together with their reciprocals.
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References
P. Batra, On small circles containing zeros and ones of analytic functions, Complex Var. Theory Appl. 49 (2004), 787–791.
L. Bieberbach, Über die Verteilung der Eins- und Nullstellen analytischer Funktionen, Math. Ann. 85 (1922), 141–148.
V. D. Blondel, R. Rupp and H. S. Shapiro, On zero and one points of analytic functions, Complex Var. Theory Appl. 28 (1995), 189–192.
V. D. Blondel, E. Sontag, M. Vidyasagar and J. C. Willems, Open Problems in Mathematical Systems and Control Theory, Springer, London, 1999.
C. Carathéodory, Sur quelques généralisations du théorème de M. Picard, C. R. Math. Acad. Sci. Paris 141 (1904), 1213–1215.
A. A. Gol’dberg, On a theorem of Landau type, Funct. Anal. Appl. (translation of Funktsional. Anal. i Prilozhen 17 (1973), 200–206.
W. K. Hayman, Some remarks on Schottky’s theorem, Proc. Cambridge Phil. Soc. 43 (1947), 442–454.
W. K. Hayman, Subharmonic Functions, vol. 2., Academic Press, London, 1989.
J. A. Hempel, Precise bounds in the Theorems of Schottky and Picard, J. Lond. Math. Soc., II. Ser. 21 (1980), 279–286.
J. A. Jenkins, On a problem of A. A. Gol’dberg, Ann. Univ. Mariae Curie-Skłodowska, Sect. A (1983), 83–86.
E. Landau, Über eine Verallgemeinerung des Picardschen Satzes, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, Berlin (1904), 1118–1133.
A. M. Ostrowski, Asymptotische Abschätzung des absoluten Betrages einer Funktion, die die Werte 0 und 1 nicht annimmt, Comment. Math. Helvet. 5 (1933), 55–87.
G. Pólya and G. Szegő, Aufgaben und Lehrsätze aus der Analysis I, Springer-Verlag, Berlin, 1970.
R. Rupp, A covering theorem for a composite class of analytic functions, Complex Var. Theory Appl. 25 (1994), 35–41.
F. Schottky, Über den Picard’schen Satz und die Borel’schen Ungleichungen, Sitzungs-berichte der Königlich Preussischen Akademie der Wissenschaften, Berlin (1904), 1244–1262.
S. Zhang, On explicit bounds in Schottky’s theorem, Complex Var. Theory Appl. 13 (1990), 161–171.
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Batra, P. On Gol’dberg’s Constant A2 . Comput. Methods Funct. Theory 7, 33–41 (2007). https://doi.org/10.1007/BF03321629
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DOI: https://doi.org/10.1007/BF03321629
Keywords
- Gol’dberg’s second constant
- Schottky’s Theorem
- Borel-Hadamard inequalities
- value distribution
- holomorphic functions